Prob. Sem. / Math Finance Colloq.

Parameter estimation for Gaussian

sequences: long memory motivations in

�nance, sharp asymptotic normality and

non-normality.

Frederi G. Viens, Dept. Statistics, Purdue University;

Division of Mathematical Sciences, National Science Foundation

Los Angeles, Sep 18, 2015

Plan of the talk

0. [Blackboard] Motivation from long memory stochastic volatility

1. Partially observed fractional Ornstein-Uhlenbeck processes: from

continuous to discrete observations

2. Scale parameter estimators for discretely observed general stationary

Gaussian processes

3. Optimal fourth and third moment theorems (Malliavin)

4. Speed of normal convergence in total variation for stationary Gaus-

sian power variations

5. Non-normal convergence

Joint work with

� L�eo Neufcourt, Columbia University,

� Khalifa Es-Sebaiy, B. El Onsy, Cadi Ayyad University (Morocco)

� Luis Barboza, Universidad de Costa Rica (Costa Rica)

� Work partially funded by

{ CIMFAV Universidad de Valpara��so (Chile)

{ Ecole Polytechnique, Palaiseau (France)

{ NSF

{ Fulbright Commission

1 Partially observed Gaussian processes� Classical case: OU process (Lipster-Shiryaev 78)

* dX (t) = ��X (t) dt+dV (t) ; t 2 [0; T ]; increas, horiz. T !1* explicit non-stationary or stationary solutionR t0 e��(t�s)dV (s) or

R t�1 e

��(t�s)dV (s) no matter what V is.

� QUESTION: estimate � > 0 .

� ANSWERS: when V is Brownian motion, since d hV i = dt* using Girsanov, get MAXIMUM LIKELIHOOD ESTIMATOR

�T = �R T0 X (s) dX (s)R T0 jX (s)j2 ds

:

* another interpretation: (ORDINARY) LEAST SQUARES ESTIM

indeed, interpret Ito integralR T0 X (s) dX (s) =

R T0 X (s) _X (s) ds

then minimize L2 (ds)-norm of the regression error

� 7!Z T0

��� _X (s) + �X (s)���2 ds:

� OU process driven by an OU process :

* Replace V with another OU process:

* dV (t) = ��X (t) dt+ dB (t)

* DIFFICULTY: ONLY OBSERVE X BUT ESTIMATE (�; �).

* Bravely and na��vely assume �T still works

* Observed version of V : V (t) = X (t) + �TR t0X (s) ds

* Bravely and na��vely try

�T =

R T0 V (s) dV (s)R T0

���V (s)���2 ds :* Bercu Proia and Savy SPL 2014 proved this intuition is incorrect:

limT!1

��T ; �T

�=

� + �;

�� (� + �)

(� + �)2 + ��

!:

� OU process driven by a fractional OU process (H > 1=2):

* Same situation as above, but B is fractional Brownian motion.

* First look at fully observed case: MLE and LSE do not coincide

{ MLE (Kleptsyna and Le Breton SISP 2002)

{ LSE (Nualart and Hu SPL 2010)

they propose � but stoch. integ. is of Skorohod type

they propose an unrelated method of moments

* We adapt method of Bercu Proia and Savy:

{ need the Malliavin calculus to recompute all needed asymptotics;

{ evaluate Skorohod integs by converting to Young integs;

{ invoke the Nualart-Peccati criterion to prove asymp. normality

limT!1

��T ; �T

�=

0BB@� + �; �� (� + �)

(� + �)2 + �2�2H��2�2H��2H���2H

1CCA =: (��; ��) a.s.;limT!1

pT��T � ��; �T � ��

�= N (0;�) with � explicit, H < 3=4:

� Same problem, now assume data is discrete: timestep �n,* Generic quadratic variation Qn (Z) :=

1n

Pnk=1Z (k�n).

* ReplaceR T0 jX (s)j2 ds by Qn (X).

* Also need to discretize �t :=R t0X (s) ds with �k�n = �n

Pki=1X (i�n).

* ReplaceR T0

���V (s)���2 ds by Qn (X) + �����T ���2Qn ���* to de�ne

���; ��

�as solution to F

���; ��

�=�Qn (X) ; Qn

����where

F (x; y) = H�(2H)�

8<:1

y2�x2�y2�2H � x2�2H ; x�2H � y�2H

�if x 6= y�

(1�H)x�2H ; Hx�2H�2�

if x = y:

* We prove a.s. cvce via control of errors from continuous case;

need intermediate estimator based on Qn (�);

need sampling rate �n � n�1

1+H�":

* We prove CLT based on a.s. control of errors, and our continuous

CLT

but need a stronger sampling rate �n = o

n� 11+2H=3

!:

2 Discretey observed Gaussian processes (in progr.)� Work directly with the discrete observations from stationary continuous-time process, and exploit 4th or 3rd moment theorem

* For quadratic estimators with stationary processes, setup is equiv-

alent to

Qn (Z) :=1

n

nXk=1

jZ (k)j2

where Z is stationary Gaussian sequence with auto-covariance �

* Let = � (0) = V ar (Z (k)); let Vn (Z) :=pn (Qn (Z)� )

* By 3rd moment theorem

dTV

0B@ Vn (Z)qV ar (Vn (Z))

; N (0; 1)

1CA ��Pn

k=1 � (k)3=2�2

�Pnk=1 � (k)

2�3=2

and in fact V ar (Vn (Z)) � n�1Pnk=1 � (k)

2 and need not converge.

* More powerful strategy: no sampling restriction: �n = 1 is OK.

� Other powers (Hermite variations): 3rd moment theorem not valid;

* optimal 4th moment theorem still holds, repeat detailed analysis;

* or just use non-optimal result of Nourdin & Peccati (PTRF, 2008):

dTV

0B@ Vn (Z)qV ar (Vn (Z))

; N (0; 1)

1CA � Cvuuuut�4

0B@ Vn (Z)qV ar (Vn (Z))

1CA:

� The actual speeds depend on how quicky V ar (Vn (Z)) stabilizes.* Ease of extension to non-stationary data.

* Have strategy to avoid H < 3=4 condition for fGn

� Application to fractional OU and OUfOU:* fOU: limQn (X) = H� (2H) �2H is a method of moments;

* OUfOU: conjec : a.s. & CLT for our previous���; ��

�with �n = 1.

� When � (k) = k2H�2L (k) with L slowly varying and H > 3=4,

* uctuations have H-Rosenblatt-distributed limit ;

* open problem to �nd optimal TV or Wasserstein speed.

3 Malliavin calculus tools

References

[NuP] Nualart, D.; Peccati, G. (2005). Central limit theorems for se-

quences of multiple stochastic integrals. Ann. Probab. 33 (1),

177-193.

[BBL] Bierm�e, H.; Bonami, A.; Le�on, J.R. (2011). Central limit theo-

rems and quadratic variations in terms of spectral density. Elec-

tron. J. Probab. 16 (13), 362-395.

[BBNP] Bierm�e, H.; Bonami, A.; Nourdin, I.; Peccati, G. (2013), Opti-

mal Berry-Esseen rates on the Wiener space: the barrier of third

and fourth cumulants. ALEA 9 (2), 473-500.

[NP] Nourdin, I.; Peccati, G. (2013). The optimal fourth moment the-

orem. In Proceedings of the Amer. Math. Soc, 2014.

Theorem 3.1 [Sharp 4th moment thm in total variation: [NuP], [NP]].

� Assumptions:

(1) (Fn)n�0 sequence in a �xed Wiener chaos;

(2) V ar [Fn] = 1

� Conclusions:

(1) [NuP] The following two statements are equivalent:

(a) (Fn)n�0 converges in law towards N (0; 1) ;(b) E[F 4n]! 3 = E[N (0; 1)4]:

(2) [NP] Let Mn := max(E[F 4n]� 3;���E[F 3n]���): If (1a) holds, then

9c; C > 0 : cMn � dTV (Fn; N) � CMn:

Where does such a result come from?

Nourdin-Peccati Analysis on Malliavin calculus with Stein's lemma.

Stein's lemma =) bound on dTV (uniform dist. of probab. measures):

Let FTV =nf : kfk1 �

q�=2 ;

f 0 1 � 2o: Then for X w/ density,

dTV (X;N (0; 1)) � supf2FTV

���E hf 0 (X)i� E [Xf (X)]��� :* Assume L2 () is w.r.t a �xed BM on [0; 1], say.

* ForX 2 D1;2 with E [X] = 0, V ar [X] = 1, letGX :=R 10 DsX Ds

��L�1

�X ds.

* Then

E [Xf (X)] = EhGXf

0 (X)i:

* Hence

dTV (X;N (0; 1)) � 2E [j1�GX j] :

* Moreover if X 2 D1;4, then

E [j1�GX j] � 2rEhj1�GX j2

i= 2

pV ar [GX] :

Special case: X 2 qth chaos.

* Then by def,��L�1

�X = q�1X and thus GX = q�1 kDXk2.

* Then work a little hard to estimatepV ar [GX] � E

hX4

i� 3. Thus

dTV (Fn; N) � CqrEhX4

i� 3:

How did [NP] get better estimate?

* NOTATION: cumulants �4 (X) := EhX4

i� 3 and �3 (X) := E

hX3

i* Iterate a polarization and iteration of GX up to order 4, to get

dTV (Fn; N) � Cq j�3j+ �4 +

s�4

��23 + �

3=24

�!� cqmax (j�3j ; �4)

* use f = sin and f = cos to �nd lower bounds with j�3j and �4respectively.

� Stationary centered Gaussian sequence: (Xn)n2ZX is jointly Gaussian and E [Xn] = 0;

9� on Z : 8k; n 2 Z, E�XnXn+k

�= � (k);

� (0) = V ar [Xn] = 1 without loss of generality;

� Only technical assumptions:� is of constant sign;

j�j decreases near +1.

� Normalized centered quadratic variation

Vn :=1pn

n�1Xk=0

�X2k � 1

�;

vn := EhV 2ni= V ar [Vn] ;

Fn :=Vnpvn:

Therefore Fn is a sequence in second Wiener chaos with EhF 2ni= 0.

� Breuer-Major theorem (special case):

if �2 :=Pk2Z � (k)

2 = limn!1 vn <1 then Vn !law

N�0; �2

�.

� Beyond this theorem, there are cases where Fn ! N�0; ~�2

�even

if �2 = 1 and law(Vn) diverges. In that case, we say that normal

convergence holds without a Breuer-Major theorem.

� Speed of Breuer-Major convergence in total variation:* Classical Berry-Ess�een (iid case): dTV (Vn;N (0; 1)) � c=

pn

* the case of fGn (increments of fBm): � (k) � H (2H � 1) k2H�2

{ Before [BBNP], it was believed that classical Berry-Ess�een rate

holds for fGn only if H < 3=5;

{ thanks to [BBNP], and [NP], we know Berry-Ess�een rate holds

if and only if H < 2=3, and other rates can be computed,

for example:

* if H = 2=3, then dTV (Fn;N (0; 1)) � n�1=2 log2 n ;* if H 2 (2=3; 3=4) then dTV (Fn;N (0; 1)) � n6H�9=2 .

4 Third moment theorem

IDEA: use [NP]'s Theorem 3.1;

NOTATION: �4 (Fn) := EhF 4ni� 3 and �3 (Fn) := E

hF 3ni

REQUIREMENT: compute �4 (Fn) and �3 (Fn) as sharply as possible.

LUCKY BREAK: don't need �4 (Fn) as sharply as �3 (Fn) :)

Proposition 4.1 [Sharp cumulants estimates]

2

v3=2npn

0@ Pjkj<n

j�(k)j3=21A2 � j�3(Fn)j � 8

v3=2npn

0@ Pjkj<n

j�(k)j3=21A2 ;(1)

�4(Fn) =const

v2n n

0@ Pjkj<n

j�(k)j4=31A3 ; (2)

n�1Pk=0

�2 (k) � vn � 2n�1Pk=0

�2 (k) : (3)

By Holder's inequality forPjkj<n j�(k)j2=3+2=3 with p = 9

4; q =95, in-

equalities (1) and (2) imply:

Corollary 4.2 [Lucky shortcut] n1=4�4(Fn)3=4 = O (j�3(Fn)j).

Consequently, by 4th moment theorem of [NuP],

�3(Fn)! 0 =) �4(Fn)� 3! 0 =) limn!1law(Fn) = N (0; 1).

And by sharp 4th moment theorem of [NP], the convergence is in total

variation, and all the converses also holds. More precisely, �4(Fn) =

o (j�3(Fn)j) and we have the following.

Theorem 4.3 [Sharp third moment theorem in TV]

(i) limn!1law(Fn) = N (0; 1).() (ii) limn!1 �3 (Fn) = 0;

() (iii) limn!1 �4 (Fn) = 0;

() (iv) "n :=�P

jkj<n j�(k)j3=2�2v�3=2n n�1=2 = o (1) :

In this case, �3 (Fn) � "n , and "n is the exact TV convergence rate:

dTV (Fn;N (0; 1)) � �3 (Fn) � "n:

5 Non-normal convergence

References

[BN] Breton, J.C.; Nourdin, I. (2008). Error bounds on the non-

normal approximation of Hermite power variations of fractional

Brownian motion. Electron. Comm. Probab. 13, 482-493.

[DM] Davydov, Y.A.; Martynova, G. V. (1987). Limit behavior of

multiple stochastic integral. Statistics and control of random

process. Preila, Nauka, Moscow, 55-57 (in Russian).

[DM79] Dobrushin, R.L.; Major, P (1979). Non-central limit theorems

for nonlinear functionals of Gaussian �elds. Z.Wahrsch. Verw.

Gebiete 50 (9), 27-52.

[T79] Taqqu, M. S. (1979). Convergence of integrated processes of

arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 (1), 53-

83.

Theorem 5.1 [Dobrushin-Major / Taqqu 79 (special case)]

Assumption: � (k) = k2H�2L (k) where L is slowly varying at 1 and

H 2 (3=4; 1).Conclusion [DM79]: Fn converges in law to the law of a Rosenblatt r.v.

F1 =ZZR2

jxyjH�1=2pKH

ei(x+y) � 1i (x+ y)

W (dx)W (dy) :

HereW is a C-valued white noise: on R+, W (dx) = B1 (dx)+iB2 (dx)

for B1; B2 iid BMs; W (�dx) =W (dx), W (dx)2 = 0, jW (dx)j2 = dx.

[BN] �nd TV speed of convergence by using a classical result [DM]:

* if Fn and F1 are on in the same Wiener chaos, and V ar [Fn] bounded

* then for large n, dTV (F; F1) � cF1�Eh(F � F1)2

i�1=4.

Case L = 0 : using self-similarity of fBm [BN] prove

dTV (F; F1) � cF1n3=4�H :

Unfortunately, we �nd this good speed only works for fGn...

� Log-modulated power spectral density:

{ let H 2 (3=4; 1) and � � 0, let L (y) = log2� (jyj) (or asymp).Let

q (x) := CH;� jxj1�2H L e�

jxj

!:

{ This q 2 L1�S1; dx

�, q is C1 except at 0; thus q � Fourier

series of its Fourier inverse �.

{ I.e. let X with covariance function

� (k) :=1

2�

Z ���q (x) cos (kx) dx

then X has spectral density q and def of � is Fourier inversion.

{ Recompute � by changing variables:

� (k) =CH;�

2�k2H�2

Z k��k�

jxj1�2H cos (x)L ke�

jxj

!dx:

Theorem 5.2 Using above X, with a c depending only on H and �,

dTV (Fn; F1) �c

log1=2 n

where, with K0 = (2�(2�2H) cos(�(1�H)))2(4H�2)(4H�3) , F1 is Rosenblatt law

F1 =ZZR2

jxyjH�1=2qK0H

ei(x+y) � 1i (x+ y)

W (dx)W (dy) :

� Technical proof:* Typically: � has long memory, thus � =2 `1 (Z), cannot invokeclassical Fourier inversion; prove it or assume it by working with q

directly.

* To apply a meta-theorem: use a trade o� between a speed of cvce

to limiting kernel and speed of integrability of cuto� kernel at 1.* need precise estimates of � and nvn :

withKH := 1�

R10 jxj1�2H cos (x) dx = 2� (2� 2H) cos (� (1�H)) ;

� (k) = CH;�KHL (k) k2H�2

1 +O

1

L (k)

!!;

nvn =�CH;�

�2K0H n4H�2 L2 (n)

1 +O

1

logn

!!: